Redirect on client side means following steps:
User request a Page using an URI,
Server sent a page based on that.
So, it Client-side -> Server-side -> Client-side -> Server-side -> Client-side.
Redirect on Server-Side means:
Client-side -> Server-side -> Client-side (headers only)* -> Server-side -> Client-side.
<script> window.location = 'http://www.mywebsite.com/new-url.php'; </script>
B. Using HTML Meta Tag, inside HEAD section:
<meta http-equiv="refresh" content="0;url=http://www.mywebsite.com/new-url.php" />
PHP/Server Side Redirect:
Redirect using PHP is done using header() function.
<?php $loc = 'http://www.mywebsite.com/new-url.php'; header("Location: $loc"); die(0); ?>
Here you have served another page content by redirecting the user to new location because you have changed the page address, the user is looking for or for some other reason. User will smoothly get the content.
Update: I have made correction. Thanks to those who have commented and pointed mistake about the server-side redirects.